The Lace Expansion and its Applications: Ecole d'Eté de Probabilités de Saint-Flour XXXIV - 2004

ISBN-13: 9783540311898 / Angielski / Miękka / 2006 / 233 str.

Gordon Slade;Jean Picard

Opis

Three series of lectures were given at the 34th Probability Summer School in Saint-Flour (July 6 24, 2004), by the Professors Cerf, Lyons and Slade. We have decided to publish these courses separately. This volume contains the course of Professor Slade. We cordially thank the author for his performance at the summer school, and for the redaction of these notes. 69 participants have attended this school. 35 of them have given a short lecture. The lists of participants and of short lectures are enclosed at the end of the volume. The Saint-Flour Probability Summer School was founded in 1971. Here are the references of Springer volumes which have been published prior to this one. All numbers refer to theLecture Notes in Mathematics series, except S-50 which refers to volume 50 of the Lecture Notes in Statistics series. 1971: vol 307 1980: vol 929 1990: vol 1527 1998: vol 1738 1973: vol 390 1981: vol 976 1991: vol 1541 1999: vol 1781 1974: vol 480 1982: vol 1097 1992: vol 1581 2000: vol 1816 1975: vol 539 1983: vol 1117 1993: vol 1608 2001: vol 1837 & 1851 1976: vol 598 1984: vol 1180 1994: vol 1648 2002: vol 1840 1977: vol 678 1985/86/87: vol 1362 & S-50 1995: vol 1690 2003: vol 1869 1978: vol 774 1988: vol 1427 1996: vol 1665 2004: vol."

Recenzje

From the reviews:

"The book consists mainly of materials for an advanced master course. ... The book focusses on the modeling of mechanics, where the concept of a model shall be understood in its broadest sense, namely as a mathematical structure which describes mechanical phenomena. ... A list of notations and an index are helpful to the reader. The little book with about 200 pages tackles many theories and concepts, without going much into detail." (Albrecht Bertram, Zentralblatt MATH, Vol. 1115 (17), 2007)

"This work is based on the author's lectures on the lace expansion and its applications ... . It contains the recent developments in the analysis of the lace expansion and the scaling limits of the critical objects discovered since the publication ... . It also contains further applications of the lace expansion to other models and an extensive list of references. ... The subject is still growing, and studying the lecture notes by Slade is a good starting point to learn the subject." (Akira Sakai, Mathematical Reviews, Issue 2007 m)

Spis treści

Simple Random Walk.- The Self-Avoiding Walk.- The Lace Expansion for the Self-Avoiding Walk.- Diagrammatic Estimates for the Self-Avoiding Walk.- Convergence for the Self-Avoiding Walk.- Further Results for the Self-Avoiding Walk.- Lattice Trees.- The Lace Expansion for Lattice Trees.- Percolation.- The Expansion for Percolation.- Results for Percolation.- Oriented Percolation.- Expansions for Oriented Percolation.- The Contact Process.- Branching Random Walk.- Integrated Super-Brownian Excursion.- Super-Brownian Motion.

O autorze

Gordon Slade is Professor at the University of British Columbia since 1999. Before he was Lecturer at the University of Virginia from 1985 to 1986 and  Professor at the McMaster University from 1986 to 1999. The Author has been awarded the UBC Killam Research Prize (Senior Science Category) in 2004 and the Prix de l'Institut Henri Poincaré--with Remco van der Hofstad--in2003. In 2003 he was Stieltjes Visiting Professor, in 1995 Coxeter-James Lecturer for the Canadian Mathematical Society. Since 2000 he is Fellow of the Royal Society of Canada. 

Blurb

The lace expansion is a powerful and flexible method for understanding the critical scaling  of several models of interest in probability, statistical mechanics,
and combinatorics, above their upper critical dimensions. These models include the self-avoiding walk, lattice trees and lattice animals, percolation, oriented percolation, and the contact process. This volume provides a unified and extensive overview of the lace expansion and its applications to these models.  Results include proofs of existence of critical exponents and construction of scaling limits. Often, the scaling limit is described in terms of super-Brownian motion.